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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1071 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32179. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1071.7 | ⊢ 𝐷 = (ω ∖ {∅}) |
Ref | Expression |
---|---|
bnj1071 | ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1071.7 | . . 3 ⊢ 𝐷 = (ω ∖ {∅}) | |
2 | 1 | bnj923 31938 | . 2 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω) |
3 | nnord 7577 | . 2 ⊢ (𝑛 ∈ ω → Ord 𝑛) | |
4 | ordfr 6199 | . 2 ⊢ (Ord 𝑛 → E Fr 𝑛) | |
5 | 2, 3, 4 | 3syl 18 | 1 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 ∅c0 4288 {csn 4557 E cep 5457 Fr wfr 5504 Ord word 6183 ωcom 7569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-om 7570 |
This theorem is referenced by: bnj1030 32156 bnj1133 32158 |
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